Bus Admittance Matrix Research Articles

A Sparse Tableau Formulation for Node-Breaker Representations in Security-Constrained Optimal Power Flow

Realistic representations of contingencies in AC optimal power flow (ACOPF) often challenge traditional bus-branch network models. Derived from nodal analysis, such approaches represent network constraints in terms of the familiar bus admittance matrix, $Y_{\text{bus}}$ . A fixed $Y_{\text{bus}}$ is unable to represent common circuit breaker actions such as bus splitting. Work-arounds for $Y_{\text{bus}}$ -based analysis typically rely on topology processing, switching between different $Y_{\text{bus}}$ matrices depending on breaker settings. In this paper, we propose a very general node-breaker approach, employing multi-port element models and using a sparse tableau formulation (STF) for network constraints. Instead of treating breaker action as altering network topology, and hence changing the structure of Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL) equations, this approach represents a breaker's position as impacting only constraints associated with a single component, thereby maintaining fixed structure in the KVL and KCL constraints. While larger numbers of variables are required, STF proves sparser than $Y_{\text{bus}}$ formulations. Numerical case studies herein demonstrate that STF provides computational efficiency comparable to a $Y_{\text{bus}}$ -based ACOPF at the scale of several hundred buses, and lower computational cost in an example of over one thousand buses.

Allocation of Static Var Compensator in Electric Power Systems Considering Different Load Levels

This paper proposes an approach to determine the optimal location of static var compensators (SVCs) in electric power systems in order to improve voltage profile and minimize active power losses. A multi-scenario framework that includes different load levels with different time periods is considered in this approach. The problem is formulated as a mixed-integer nonlinear programming problem using an optimal power flow (OPF). The SVC value and location are modeled as a variable susceptance inside the bus admittance matrix and as a binary decision variable, respectively. The problem is solved using the branch and bound algorithm associated with the OPF. Studies and simulations were conducted on the IEEE 118-bus test system considering variations in both the objective function and the amount of SVCs to be allocated. Analysis of results demonstrate that the performance of the power system can be effectively enhanced due to the optimal allocation of SVC equipment if considering different load levels with different time periods for the allocation of SVCs, rather than allocate the SVCs separately.

Modeling and Analysis of Open-Delta Step Voltage Regulators for Unbalanced Distribution Network With Photovoltaic Power Generation

Open-delta connected step voltage regulators (SVRs) have been widely installed into distribution networks, and their characteristic model (auto-transformer model) has been established over the years for power system analysis. However, these models are not fully developed to a complete bus admittance matrix form, which can be readily used for unbalanced load flow programs. Further, the debate on the neutral shift after implementation of open-delta SVRs has added difficulties on the modeling. There have been confusions on what is the neutral shift for a three-wire network without neutral and where the neutral is shifted to. Recently, accurate modeling of such a network component has become exceptionally important especially with the rapid proliferation of photovoltaic (PV), and there is a strong requirement of evaluating PV integration impacts on distribution systems. Propelled by all these necessity of modeling, this paper develops a new mathematical model of the open-delta SVR in a bus admittance matrix form which can be easily incorporated into suitable unbalanced load flow programs. Further, the neutral shift issue is also clarified during the model development, and in addition a method for tap position estimation is derived. All the established models are validated by field measurement.

Decomposed Newton algorithm-based three-phase power-flow for unbalanced radial distribution networks with distributed energy resources and electric vehicle demands

This paper proposes a three-phase power-flow algorithm using graph theory, injected current, and matrix decomposition techniques for unbalanced radial distribution networks. A decomposed Newton-Raphson (DNR) method is applied to solve the set of nonlinear power equations described in polar form. Unlike conventional Newton-Raphson-based methods, the proposed DNR algorithm does not involve calculating the lower and upper triangular matrix (LU) factorization, Gaussian elimination, and inversion of the full bus admittance matrix or Jacobian matrix and building the bus impedance matrix; it also requires less computation time and has high robustness with respect to the X/R ratio and load changes. The mathematical component models, such as three-phase conductors, transformers, automatic voltage regulators (AVRs), ZIP load demands, shunt capacitors/reactors, inverter-based distributed energy resources (DERs) and electric vehicle (EV) demands can easily be integrated into the proposed algorithm by using the injected current technique. Therefore, a three-phase power-flow problem can be decomposed into three single-phase power-flow problems with individual phase representation. To validate the performance and effectiveness of the proposed algorithm, four three-phase IEEE test systems and a practical Taiwan Power Company (Taipower) distribution system are used for comparisons. The results reveal that the proposed algorithm has good potential for improving the computational efficiency of optimal planning and design and also real-time power dispatch applications, even for ill-conditioned distribution networks.

Comprehensive Modeling of Three-Phase Distribution Systems via the Bus Admittance Matrix

The theme of this paper is three-phase distribution system modeling suitable for the Z-Bus load-flow. Detailed models of wye and delta constant-power, constant-current, and constant-impedance loads are presented. Models of transmission lines, voltage regulators, and transformers that build the bus admittance matrix (Y-Bus) are laid out. The Z-Bus load-flow is then reviewed and the singularity of the Y-Bus in case of certain transformer connections is rigorously discussed. Based on realistic assumptions and conventional modifications, the invertibility of the Y-Bus is proved. Last but not least, the MATLAB scripts that construct the detailed component models for the IEEE 37-bus, IEEE 123-bus, and 8500-node feeders as well as the European 906-bus low-voltage feeder are provided.

A new approach for evaluation of the bus admittance matrix from synchrophasors: (A statistical Ybus estimation approach)

In this research, a new parameter estimation technique for determining the bus admittance matrix (Ybus) is proposed to further calibrate power system models. Ybus estimation is done using recorded PMU synchrophasor measurements. The approach proposed in this research is based on recognizing that bus injection currents Ibus can be viewed as signals produced by a random process. In this manner, the corresponding bus voltages Vbus are also stochastic signals that are related through a cross-covariance matrix to the vector Ibus. Using estimation techniques developed for statistical signal processing, the cross-covariance matrix is shown to be Zbus. Research done previously in this area requires the prior knowledge of the electrical interconnection as well as full availability of PMU measurements. This method does not depend on prior knowledge of the electrical interconnection between the buses. It extracts the electrical interconnection and the parameters of the network model in the form of Ybus. It also does not require full availability of PMU measurements at each bus in the system. It estimates the partial Ybus based on the available PMUs.

A new sequential AC–DC power flow algorithm for multi‐terminal HVDC systems

This paper presents a new algorithm based on the sequential method for power flow calculation in integrated multi‐terminal, high‐voltage, direct current (HVDC) systems. Unlike similar studies in the literature, a real equivalent circuit model is considered for under‐load tap changer (ULTC) transformers of the DC converters, for the first time. So, new DC equations are obtained. Thus, exact and accurate results can be obtained for practical applications by the proposed algorithm. Adjustment effects of the DC converters' ULTCs tap values are included in the Jacobian matrix instead of the bus admittance matrix in the sequential AC power flow algorithm as well as other ULTCs in AC system. To this aim, new equations for the calculation of power and Jacobian matrix elements are obtained for the AC system. The proposed approach is tested on the modified IEEE 17‐bus AC–DC test system. Numerical results show that the proposed approach is accurate and reliable in convergence. © 2017 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Transmission line fault detection and location using Wide Area Measurements

Knowledge of both voltage and current data is conventionally entailed for locating fault on transmission network. However, owing to the non-linearity of the system, the process becomes iterative and cumbersome. In order to address this issue, a new technique for fault location on transmission lines using only voltage measurements obtained from Wide Area Measurement Systems (WAMS) and the network bus admittance matrix has been reported in this paper. It is based on the fundamental law of electrical networks. It delineates a single step method to find out the faulted line and the exact location of fault. The decision is based on change in injected current at various buses following the line fault. The method has been tested on IEEE 14 bus and New England 39 bus system. Applicability of the proposed algorithm is independent of fault type, location of fault and fault resistance. This method not only proves to be accurate but also furnishes quick results with reduced computational burden. Besides, the faulted phase as well as fault type detection has been suggested in this paper using same measurements.

A new sequential power flow algorithm for AC/DC systems including independent multiterminal DC subsystems

This paper presents a new algorithm based on the sequential method for power flow calculation in multiterminal AC/DC systems. The proposed approach differs from other similar studies in the literature by handling two main properties. Firstly, a real equivalent circuit model is considered for the under-load tap changer transformers (ULTCs) of DC converters for the first time in the literature. Hence, new DC equations are obtained. Thus, exact and accurate results can be obtained for practical applications using the proposed algorithm. Tap values adjustment effects of the ULTCs connected to DC converters are included into the Jacobian matrix instead of the bus admittance matrix in the sequential AC power flow algorithm, as well as other ULTCs in the AC system. For this aim, new equations for calculating power and Jacobian matrix elements are obtained. Secondly, more than one independent multiterminal DC subsystem in terms of DC connection is included in the study. DC power flow calculation is performed for each DC subsystem independently in the algorithm. Thus, modularity and flexibility are provided for including or removing any DC subsystems without needing to change the proposed algorithm. The proposed approach has been tested on a modified IEEE 20-bus AC/DC test system. The proposed AC/DC power flow algorithm was written and performed in MATLAB. The results show that the proposed approach is accurate and reliable in convergence.

Three-Phase Unbalanced Power Flow Using a <formula formulatype="inline"> <tex Notation="TeX">$\pi$</tex> </formula>-Model of Controllable AC-DC Converters

Microgrids are unique in that they can combine unbalanced three-phase systems with other AC and DC network sections, which may include a range of renewable energy sources, energy storage elements, and controllable AC-DC converters. Existing unbalanced power flow techniques such as the Ladder Iterative Technique and the three-phase Newton-Raphson (NR) method can analyze microgrids in sections but lack a complete system representation. Hence, there is a need for a power flow algorithm that considers the complete system model and solves it. A novel $\pi$ -model of a controllable AC-DC converter and a single set of power balance equations for modeling a grid comprising multiple three-phase AC and DC sections is proposed. The $\pi$ -model of a controllable AC-DC converter enables its inclusion into the network bus admittance matrix (YBUS) along with three-phase AC and DC network sections. Verification of the $\pi$ -model is also described in the paper. The results of power flow studies with three-phase balanced and unbalanced AC and DC network sections are presented. The outcome of the $\pi$ -model verification study and power flow study show that the proposed $\pi$ -model is consistent and accurate. While the proposed model was developed for microgrids, it is applicable for all power system analysis applications.

Modelling of TCSC for voltage regulation in power systems

Purpose The purpose of this paper is to propose a voltage regulation solution in power systems through adjusting the power flow of the system via thyristor controlled series compensator (TCSC). For this aim, a new power flow model has been proposed based on TCSC. Design/methodology/approach TCSC’s admittance effect has been included as state variable into the Jacobian matrix to avoid divergence problem. TCSC’s admittance is ignored in the bus admittance matrix to prevent rebuilding requirement of the bus admittance matrix in each power flow iteration. So, faster convergence for power flow calculation has been provided. For this aim, new power equations have been obtained. Also, the proposed approach has not required to handle each terminal of TCSC as an individual bus in the power flow calculation. So, increasing of the Jacobian and bus admittance matrixes caused by the total bus number has been prevented. Findings The proposed approach has been tested on IEEE 57-bus test system. The obtained results have proved that the proposed approach has provided efficient, reliable and fast convergence. Originality/value This study is the first one that uses TCSC for voltage regulation in the literature. On the other hand, the results have shown that the approach of considering the TCSC admittance values as state variables provides robust convergence, according to the approaches that consider TCSC firing angles as state variables.

The Formulation of a Power Flow Using <inline-formula> <tex-math notation="LaTeX">$d-q$</tex-math> </inline-formula> Reference Frame Components—Part I: Balanced <inline-formula> <tex-math notation="LaTeX">$3\phi$ </tex-math> </inline-formula> Systems

This paper develops a new approach for formulating the power flow for power systems. The developed approach is based on expressing the active and reactive power injections at each bus in a power system using the d-q-axis components of bus voltages and admittance matrix. The new power flow formulation produces a scaled Jacobian matrix, which can offer a fast convergence to the solution. The proposed power flow formulation can also model bus type conversions without affecting its accuracy or fast convergence. In addition, the d-q-axis power flow (DQPF) offers a simplified and reliable representation of photovoltaic buses that have distributed generation units (DGUs). The DQPF is implemented with a step-by-step procedure for performance evaluation on several power systems operated at different conditions. Performance results demonstrate fast convergence, reduced computations, and minor sensitivity to the number of buses, loading conditions, and levels of DGU penetration. Furthermore, performance results show that DQPF power flow can attain solutions in less iterations than Newton-Raphson, Fast Decoupled, and Iwamoto power flow methods.

A robust power flow model for active power flow control via thyristor‐controlled series compensator

In this paper, a new power flow model for active power flow control through a thyristor‐controlled series compensator (TCSC) in an AC network system is proposed. The proposed power flow model is based on the Newton–Raphson method. In this model, TCSC's admittance effect is included as a state variable into the Jacobian matrix to avoid the divergence problem. Unlike similar studies in the literature, TCSC's admittance is ignored in the bus admittance matrix, and the need for rebuilding the bus admittance matrix in each power flow iteration caused by the change of TCSC's admittance is prevented. So, faster convergence for power flow calculation is achieved. For this aim, new power equations are obtained. Also, in the proposed approach, we need not consider each terminal of TCSC as an individual bus in the power flow calculation. Thus, increasing the Jacobian and bus admittance matrixes sizes caused by the total bus number is prevented. The proposed approach is tested on an IEEE 57‐bus test system. The obtained results prove that this approach provides efficient, reliable, and fast convergence. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Visualizing voltage relationships using the unity row summation and real valued properties of the FLG matrix

By manipulating the bus admittance matrix of a power system, a useful submatrix, FLG, can be derived. This matrix identifies, for every load bus, the set of generators that establish its no-load voltage, and the varying degree of their influence. The first contribution of the present work is to rigorously prove two observed properties of the FLG matrix; that it is substantially real-valued, and that its rows sum close to one. Six test systems are used in this work to validate these properties. With this proof in hand, this work also introduces a new conception of voltage profile monitoring in power systems, by explicitly mapping the relationships between load and generator voltages. This new visualization makes it easier to identify how influential each generator is in establishing the network's voltage profile. Poorly supported load buses, which may be vulnerable to voltage deviations, are clearly identified. This new visualization framework is suitable for pedagogy, research, and control room applications.

Analytical Approximations of Critical Clearing Time for Parametric Analysis of Power System Transient Stability

An analytic approximation for the critical clearing time (CCT) metric is derived from direct methods for power system stability. The formula has been designed to incorporate as many features of transient stability analysis as possible such as different fault locations and different post-fault network states. The purpose of this metric is to analyze trends in stability (in terms of CCT) of power systems under the variation of a system parameter. The performance of this metric to measure stability trends is demonstrated on an aggregated power network, the so-called two machine infinite bus network, by varying load parameters in the full bus admittance matrix using numerical continuation. The metric is compared to two other expressions for the CCT which incorporate additional nonlinearities present in the model.

A Sensitivity Approach to Model Local Voltage Controllers in Distribution Networks

Local controllers are essential in distribution networks; they are employed in classical devices such as load tap-changing (LTC) transformers and switchable shunt capacitors, and more recently in distributed generation (DG). The effective use of distribution management system (DMS) applications requires an accurate model of the interaction between the local controllers through the distribution system. This paper presents a new sensitivity matrix approach for modeling such interactions, and demonstrates its application in the implicit ZBus Gauss method for power flow computation. The sensitivity method models both PV buses (for the connection of DG) and tap position adjustments through current source injections, and consequently avoids re-factorization of the network bus admittance matrix. Numerical results on distribution networks with up to 3145 buses show that the sensitivity-based power flow method for simulating the operation of local controllers is superior to a sequential control action adjustment approach previously proposed in the literature, and that its computing time is commensurate with the performance requirements in real-time DMS applications.

Node Ordering Scheme of Large Scale Power Systems Using Sparse Matrix Techniques

Power flow is the basic tool for power system analysis which reveals the system operation in a steadystate mode for evaluation of the power system planning and operations. The accuracy, simulation time, computer storage size and convergence of any model used depend largely on the size of the bus admittance matrix of the system under study. This paper, therefore, presents the study of the bus admittance matrix of the different systems with sparse techniques. And also analyze the number of zeros and non-zeros element in the different systems with minimum ordering schemes. The proposed method is validated using a 5-bus, 30-bus, 118-bus and 300-bus systems. The results are presented in graphical form and discussed. The sparse Matrix techniques show that as the system is increasing in size, the percentage of stored bus admittance elements decreases and changing the order of the nodes gives more impact on the size of the system. Thus, an appreciable reduction in the computer memory required to store the bus admittance matrix and in turn reduces the overall simulation time.

Functional Efficacy of Omu-Aran 132/33kV Electricity Transmission Substation-Network Arrangement

This is one in a series of need driven studies to evaluate the efficacy of power delivery of radial 132/33 kV substation-network arrangements under prevailing load conditions, which represents communities expansions and social developments. The specific condition of Omu-Aran 132/33 kV substation-network arrangement (SNA), which supplies electricity to 5 load-communities in 4 respective States with similar demand cultures, was examined. Outgoing 33kV Network lines (NLs) were designated according to the names of their respective termini-cities. The description of network elements obtained from the network control centres, and network parameters were evaluated using standard formulas. From time series logged data (2007-2010) at the SNA, the maximum power statistics were evaluated to define the peak load conditions on the NLs. The SNA was modelled mathematically for load flow computational experiments using, predominantly, Bus Admittance Matrix method. Efficacy of power delivery was analysed using, mainly, voltage and thermal capacity criteria. An analysis of the results established substantial and inadmissible voltage and power losses between substation and load-communities on the existing 33 kV network lines; efficient normal mode use of these lines requires implementation of drastic load shedding, network reconstruction or reinforcement by voltage and power saving requirements.

Generalized Dynamic VSC MTDC Model for Power System Stability Studies

In this paper, a new general voltage source converter high voltage direct current (VSC MTDC) model is derived mathematically. The full system model consists of the converter and its controllers, DC circuit equations, and coupling equations. The main contribution of the new model is its valid for every possible topology of the DC circuit. Practical implementation of the model in power system stability software is discussed in detail. The generalized DC equations can all be expressed in terms of matrices that are byproducts of the construction of the DC bus admittance matrix. Initialization, switching actions resulting in different topologies and simulation of the loss of DC lines amount to a simple calculation or recalculation of the DC bus admittance matrix. The model is implemented in Matlab. Examples on a two- and six-terminal system show that the new model is indeed capable of accurately simulating VSC MTDC systems with arbitrary topology.

An improved model for structural vulnerability analysis of power networks

Electric power networks have been studied as a typical example of real-world complex networks. Traditional models for structural vulnerability analysis appear to be all based on physical topological structure. In this paper, we depict a typical power network as a weighted graph based on electrical topology by introducing its bus admittance matrix, which embodies the important characteristics of power networks in a much more realistic structure. Furthermore, the numerical simulation for both the traditional dynamical model and the proposed electrical topological model are investigated based on the IEEE 300 bus system respectively. The comparison demonstrates that the improved model is more precise and highly efficient for the analysis of structural vulnerability of power networks.